Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation. For each $\vect{v}$, define the pre-image of $\vect{v}$ to be the subset of $U$ given by \begin{equation*} \preimage{T}{\vect{v}}=\setparts{\vect{u}\in U}{\lt{T}{\vect{u}}=\vect{v}} \end{equation*}